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Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 21

Current and Direct Current Circuits - all with Video Answers

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Chapter Questions

01:59

Problem 1

In a particular cathode-ray tube, the measured beam current is $30.0 \mu$ A. How many electrons strike the tube screen every 40.0 s?

Manish Kumar
Manish Kumar
Numerade Educator
02:12

Problem 2

Suppose the current in a conductor decreases exponentially with time according to the equation $I(t)=I_{0} e^{-t / \tau}$, where $I_{0}$ is the initial current (at $t=0$ ) and $\tau$ is a constant having dimensions of time. Consider a fixed observation point within the conductor. (a) How much charge passes this point between $t=0$ and $t=\tau ?$ (b) How much charge passes this point between $t=0$ and $t=10 \tau ?$ (c) What If? How much charge passes this point between $t=0$ and $t=\infty ?$

Kayla Gephart
Kayla Gephart
Numerade Educator
02:26

Problem 3

The quantity of charge $q$ (in coulombs) that has passed through a surface of area $2.00 \mathrm{cm}^{2}$ varies with time according to the equation $q=4 t^{3}+5 t+6,$ where $t$ is in seconds. What is the instantaneous current through the surface at $t=1.00 \mathrm{s} ?(\mathrm{b})$ What is the value of the current density?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
03:04

Problem 4

A small sphere that carries a charge $q$ is whirled in a circle at the end of an insulating string. The angular frequency of revolution is $\omega .$ What average current does this revolving charge represent?

Andrija Isakov
Andrija Isakov
Numerade Educator
03:08

Problem 5

The electron beam emerging from a certain high-energy electron accelerator has a circular cross section of radius $1.00 \mathrm{mm} .$ (a) The beam current is $8.00 \mu$ A. Find the current density in the beam assuming it is uniform
throughout.(b) The speed of the electrons is so close to the speed of light that their speed can be taken as $300 \mathrm{Mm} / \mathrm{s}$ with negligible error. Find the electron density in the beam. (c) Over what time interval does Avogadro's number of electrons emerge from the accelerator?

Kayla Gephart
Kayla Gephart
Numerade Educator
05:13

Problem 6

Figure $\mathrm{P} 21.6$ represents a section of a conductor of nonuniform diameter carrying a current of $I=5.00 \mathrm{A}$. The radius of cross-sec$\operatorname{tion} A_{1}$ is $r_{1}=0.400 \mathrm{cm}$
(a) What is the magnitude of the current density across $A_{1}$ ? The radius $r_{2}$ at $A_{2}$ is larger than the radius $r_{1}$ at $A_{1}$. (b) Is the current a $A_{2}$ larger, smaller, or the same? (c) Is the current density at
$A_{2}$ larger, smaller, or the same? Assume $A_{2}=4 A_{1} .$ Specify the (d) radius, (e) current, and (f) current density at $A_{2}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
00:54

Problem 7

An aluminum wire having a cross-sectional area equal to $4.00 \times 10^{-6} \mathrm{m}^{2}$ carries a current of $5.00 \mathrm{A}$. The density of aluminum is $2.70 \mathrm{g} / \mathrm{cm}^{3}$. Assume each aluminum atom supplies one conduction electron per atom. Find the drift speed of the electrons in the wire.

Kayla Gephart
Kayla Gephart
Numerade Educator
00:49

Problem 8

A 0.900 -V potential difference is maintained across a 1.50 -m length of tungsten wire that has a cross-sectional area of $0.600 \mathrm{mm}^{2} .$ What is the current in the wire?

Kayla Gephart
Kayla Gephart
Numerade Educator
01:29

Problem 9

An aluminum wire with a diameter of $0.100 \mathrm{mm}$ has a uniform electric field of $0.200 \mathrm{V} / \mathrm{m}$ imposed along its entire length. The temperature of the wire is $50.0^{\circ} \mathrm{C}$. Assume one free electron per atom. (a) Use the information in Table 21.1 to determine the resistivity of aluminum at this temperature. (b) What is the current density in the wire? (c) What is the total current in the wire? (d) What is the drift speed of the conduction electrons? (e) What potential difference must exist between the ends of a 2.00 -m length of the wire to produce the stated electric field?

Dominador Tan
Dominador Tan
Numerade Educator
01:04

Problem 10

A lightbulb has a resistance of $240 \Omega$ when operating with a potential difference of $120 \mathrm{V}$ across it. What is the current in the lightbulb?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:33

Problem 11

Suppose you wish to fabricate a uniform wire out of $1.00 \mathrm{g}$ of copper. If the wire is to have a resistance of $R=$ $0.500 \Omega$ and all the copper is to be used, what must be (a) the length and (b) the diameter of this wire?

Kayla Gephart
Kayla Gephart
Numerade Educator
02:36

Problem 12

Suppose you wish to fabricate a uniform wire from a mass $m$ of a metal with density $\rho_{m}$ and resistivity $\rho .$ If the wire is to have a resistance of $R$ and all the metal is to be used, what must be (a) the length and (b) the diameter of this wire?

Kayla Gephart
Kayla Gephart
Numerade Educator
00:06

Problem 13

While taking photographs in Death Valley on a day when the temperature is $58.0^{\circ} \mathrm{C},$ Bill Hiker finds that a certain voltage applied to a copper wire produces a current of 1.000 A. Bill then travels to Antarctica and applies the same voltage to the same wire. What current does he register there if the temperature is $-88.0^{\circ} \mathrm{C}$ ? Assume that no change occurs in the wire's shape and size.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:21

Problem 14

An aluminum rod has a resistance of $1.234 \Omega$ at $20.0^{\circ} \mathrm{C} .$ Calculate the resistance of the rod at $120^{\circ} \mathrm{C}$ by accounting for the changes in both the resistivity and the dimensions of the rod. The coefficient of linear expansion for aluminum is $24.0 \times 10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1}$.

Dominador Tan
Dominador Tan
Numerade Educator
03:22

Problem 15

If the current carried by a conductor is doubled, what happens to (a) the charge carrier density, (b) the current density, (c) the electron drift velocity, and (d) the average time interval between collisions?

Andrija Isakov
Andrija Isakov
Numerade Educator
03:12

Problem 16

An iron wire has a cross-sectional area equal to $5.00 \times 10^{-6} \mathrm{m}^{2} .$ Carry out the following steps to determine the drift speed of the conduction electrons in the wire if it carries a current of 30.0 A. (a) How many kilograms are there in 1.00 mole of iron? (b) Starting with the density of iron and the result of part (a), compute the molar density of iron (the number of moles of iron per cubic meter). (c) Calculate the number density of iron atoms using Avogadro's number. (d) Obtain the number density of conduction electrons given that there are two conduction electrons per iron atom. (e) Calculate the drift speed of conduction electrons in this wire.

Kayla Gephart
Kayla Gephart
Numerade Educator
02:21

Problem 17

If the magnitude of the drift velocity of free electrons in a copper wire is $7.84 \times 10^{-4} \mathrm{m} / \mathrm{s}$, what is the electric field in the conductor?

Kayla Gephart
Kayla Gephart
Numerade Educator
00:27

Problem 18

The potential difference across a resting neuron in the human body is about $75.0 \mathrm{mV}$ and carries a current of about $0.200 \mathrm{mA} .$ How much power does the neuron release?

Salamat Ali
Salamat Ali
Numerade Educator
02:34

Problem 19

In a hydroelectric installation, a turbine delivers 1500 hp to a generator, which in turn transfers $80.0 \%$ of the mechanical energy out by electrical transmission. Under these conditions, what current does the generator deliver at a terminal potential difference of $2000 \mathrm{V} ?$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:56

Problem 20

Residential building codes typically require the use of 12-gauge copper wire (diameter 0.205 $\mathrm{cm}$ ) for wiring receptacles. Such circuits carry currents as large as 20.0 A. If a wire of smaller diameter (with a higher gauge number) carried that much current, the wire could rise to a high temperature and cause a fire. (a) Calculate the rate at which internal energy is produced in $1.00 \mathrm{m}$ of 12 -gauge copper wire carrying 20.0 A. (b) What If ? Repeat the calculation for a 12 -gauge aluminum wire. (c) Explain whether a 12 -gauge aluminum wire would be as safe as a copper wire.

Kayla Gephart
Kayla Gephart
Numerade Educator
03:53

Problem 21

A certain toaster has a heating element made of Nichrome wire. When the toaster is first connected to a $120-\mathrm{V}$ source (and the wire is at a temperature of $20.0^{\circ} \mathrm{C}$ ), the initial current is 1.80 A. The current decreases as the heating element warms up. When the toaster reaches its final operating temperature, the current is 1.53 A. (a) Find the power delivered to the toaster when it is at its operating temperature. (b) What is the final temperature of the heating element?

Vishal Gupta
Vishal Gupta
Numerade Educator
04:48

Problem 22

Why is the following situation impossible? A politician is decrying wasteful uses of energy and decides to focus on energy used to operate plug-in electric clocks in the United States. He estimates there are 270 million of these clocks, approximately one clock for each person in the population. The clocks transform energy taken in by electrical transmission at the average rate 2.50 W. The politician gives a speech in which he complains that, at today's electrical rates, the nation is losing 100 million dollar every year to operate these clocks.

Jonathan Ibarra
Jonathan Ibarra
Numerade Educator
03:12

Problem 23

An 11.0 -W energy-efficient fluorescent lightbulb is designed to produce the same illumination as a conventional $40.0-\mathrm{W}$ incandescent lightbulb. Assuming a cost of 0.110 dollar $/ \mathrm{kWh}$ for energy from the electric company, how much money does the user of the energy-efficient bulb save during 100 h of use?

Andrija Isakov
Andrija Isakov
Numerade Educator
01:13

Problem 24

Make an order-of-magnitude estimate of the cost of one person's routine use of a handheld hair dryer for 1 year. If you do not use a hair dryer yourself, observe or interview someone who does. State the quantities you estimate and their values.

Kayla Gephart
Kayla Gephart
Numerade Educator
02:33

Problem 25

A 100 -W lightbulb connected to a 120 -V source experiences a voltage surge that produces $140 \mathrm{V}$ for a moment. By what percentage does its power output increase? Assume its resistance does not change.

Kayla Gephart
Kayla Gephart
Numerade Educator
03:26

Problem 26

The cost of energy delivered to residences by electrical transmission varies from 0.070 dolllar$/ \mathrm{kWh}$ to 0.258 dollar$/ \mathrm{kWh}$ throughout the United States; 0.110 dollar$/ \mathrm{kWh}$ is the average value. At this average price, calculate the cost of (a) leaving a 40.0 -W porch light on for two weeks while you are on vacation, (b) making a piece of dark toast in 3.00 min with a 970 -W toaster, and (c) drying a load of clothes in 40.0 min in a $5.20 \times 10^{3}-\mathrm{W}$ dryer.

Andrija Isakov
Andrija Isakov
Numerade Educator
02:11

Problem 27

Assuming the cost of energy from the electric company is 0.110 dollar$/ \mathrm{kWh}$, compute the cost per day of operating a lamp that draws a current of 1.70 A from a 110 -V line.

Andrija Isakov
Andrija Isakov
Numerade Educator
08:22

Problem 28

An office worker uses an immersion heater to warm $250 \mathrm{g}$ of water in a light, covered, insulated cup from $20.0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ in 4.00 min. The heater is a Nichrome resistance wire connected to a 120 -V power supply. Assume the wire is at $100^{\circ} \mathrm{C}$ throughout the 4.00 -min time interval. Specify a relationship between a diameter and a length that the wire can have. (b) Can it be made from less than $0.500 \mathrm{cm}^{3}$ of Nichrome?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:27

Problem 29

A toaster is rated at $600 \mathrm{W}$ when connected to a $120-\mathrm{V}$ source. What current does the toaster carry, and what is its resistance?

Andrija Isakov
Andrija Isakov
Numerade Educator
01:54

Problem 30

A rechargeable battery of mass $15.0 \mathrm{g}$ delivers an average current of $18.0 \mathrm{mA}$ to a portable DVD player at $1.60 \mathrm{V}$ for $2.40 \mathrm{h}$ before the battery must be recharged. The recharger maintains a potential difference of $2.30 \mathrm{V}$ across the battery and delivers a charging current of 13.5 mA for $4.20 \mathrm{h} .(\mathrm{a})$ What is the efficiency of the battery as an energy storage device? (b) How much internal energy is produced in the battery during one charge-discharge cycle? (c) If the battery is surrounded by ideal thermal insulation and has an effective specific heat of $975 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$, by how much will its temperature increase during the cycle?

Dominador Tan
Dominador Tan
Numerade Educator
01:24

Problem 31

An all-electric car (not a hybrid) is designed to run from a bank of 12.0 -V batteries with total energy storage of $2.00 \times 10^{7} \mathrm{J} .$ If the electric motor draws $8.00 \mathrm{kW}$ as the car moves at a steady speed of $20.0 \mathrm{m} / \mathrm{s}$, (a) what is the current delivered to the motor? (b) How far can the car travel before it is "out of juice"?

Kayla Gephart
Kayla Gephart
Numerade Educator
02:05

Problem 32

A well-insulated electric water heater warms $109 \mathrm{kg}$ of water from $20.0^{\circ} \mathrm{C}$ to $49.0^{\circ} \mathrm{C}$ in 25.0 min. Find the resistance of its heating element, which is connected across a 240 -V potential difference.

Kayla Gephart
Kayla Gephart
Numerade Educator
01:33

Problem 33

A battery has an emf of $15.0 \mathrm{V}$. The terminal voltage of the battery is $11.6 \mathrm{V}$ when it is delivering $20.0 \mathrm{W}$ of power to an external load resistor $R$. (a) What is the value of $R$ ?
(b) What is the internal resistance of the battery?

Narayan Hari
Narayan Hari
Numerade Educator
02:44

Problem 34

Two 1.50 -V batteries $-$ with their positive terminals in the same direction-are inserted in series into a flashlight. One battery has an internal resistance of $0.255 \Omega$, and the other has an internal resistance of $0.153 \Omega .$ When the switch is closed, the bulb carries a current of $600 \mathrm{mA}$. (a) What is the bulb's resistance? (b) What fraction of the chemical energy transformed appears as internal energy in the batteries?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:28

Problem 35

An automobile battery has an emf of $12.6 \mathrm{V}$ and an internal resistance of $0.0800 \Omega$. The headlights together have an equivalent resistance of $5.00 \Omega$ (assumed constant). What is the potential difference across the headlight bulbs (a) when they are the only load on the battery and (b) when the starter motor is operated, requiring an additional 35.0 A from the battery?

Chris Johnson
Chris Johnson
Numerade Educator
03:53

Problem 36

For the purpose of measuring the electric resistance of shoes through the body of the wearer standing on a metal ground plate, the American National Standards Institute (ANSI) specifies the circuit shown in Figure P21.36. The potential difference $\Delta V$ across the 1.00 -M $\Omega$ resistor is measured with an ideal voltmeter. (a) Show that the resistance of the footwear is $R_{\text {shoes }}=\frac{50.0 \mathrm{V}-\Delta V}{\Delta V}$
(b) In a medical test, a current through the human body should not exceed $150 \mu$ A. Can the current delivered by the ANSI-specified circuit exceed $150 \mu \mathrm{A}$ ? To decide, consider a person standing barefoot on the ground plate.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:33

Problem 37

(a) Find the equivalent resistance between points $a$ and $b$ in Figure $\mathrm{P} 21.37$. (b) $\mathrm{A}$ potential difference of $34.0 \mathrm{V}$ is applied between points $a$ and $b .$ Calculate the current in each resistor.

Shoukat Ali
Shoukat Ali
Other Schools
02:26

Problem 38

Why is the following situation impossible ? A technician is testing a circuit that contains a resistance $R$. He realizes that a better design for the circuit would include a resistance $\frac{7}{3} R$ rather than $R .$ He has three additional resistors, each with resistance $R .$ By combining these additional resistors in a certain combination that is then placed in series with the original resistor, he achieves the desired resistance.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
11:11

Problem 39

Consider the circuit shown in Figure P21.39. Find (a) the current in the $20.0-\Omega$ resistor and (b) the potential difference between points $a$ and $b$.

Chris Johnson
Chris Johnson
Numerade Educator
03:06

Problem 40

Four resistors are connected to a battery as shown in Figure P21.40. The current in the battery is $I$, the battery emf is $\boldsymbol{\varepsilon},$ and the resistor values are $R_{1}=R, R_{2}=$ $2 R, R_{3}=4 R,$ and $R_{4}=3 R$ (a) Rank the resistors according to the potential difference across them, from largest to smallest. Note any cases of equal potential differences. (b) Determine the potential difference across each resistor in terms of $\boldsymbol{\varepsilon}$. (c) Rank the resistors according to the current in them, from largest to smallest. Note any cases of equal currents. (d) Determine the current in each resistor in terms of $I$. (e) If $R_{3}$ is increased, what happens to the current in each of the resistors? (f) In the limit that $R_{3} \rightarrow \infty$, what are the new values of the current in each resistor in terms of $I$, the original current in the battery?

Dominador Tan
Dominador Tan
Numerade Educator
06:56

Problem 41

Three $100-\Omega$ resistors are connected as shown in Figure P21.41. The maximum power that can safely be delivered to any one resistor is $25.0 \mathrm{W}$. (a) What is the maximum potential difference that can be applied to the terminals $a$ and $b$ ? (b) For the voltage determined in part (a), what is the power delivered to each resistor? (c) What is the total power delivered to the combination of resistors?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:14

Problem 42

A young man owns a canister vacuum cleaner marked $" 535 \mathrm{W}[\text { at }] 120 \mathrm{V}^{\prime \prime}$ and a Volkswagen Beetle, which he wishes to clean. He parks the car in his apartment parking lot and uses an inexpensive extension cord $15.0 \mathrm{m}$ long to plug in the vacuum cleaner. You may assume the cleaner has constant resistance. (a) If the resistance of each of the two conductors in the extension cord is $0.900 \Omega$, what is the actual power delivered to the cleaner? (b) If instead the power is to be at least $525 \mathrm{W}$, what must be the diameter of each of two identical copper conductors in the cord he buys? (c) Repeat part (b) assuming the power is to be at least $532 \mathrm{W}$.

Dominador Tan
Dominador Tan
Numerade Educator
04:39

Problem 43

Calculate the power delivered to each resistor in the circuit shown in Figure P21.43.

Vishal Gupta
Vishal Gupta
Numerade Educator
04:59

Problem 44

A lightbulb marked "75 W @ 120 V" is screwed into a socket at one end of a long extension cord, in which each of the two conductors has resistance $0.800 \Omega$. The other end of the extension cord is plugged into a 120 -V outlet.
(a) Explain why the actual power delivered to the lightbulb cannot be $75 \mathrm{W}$ in this situation. (b) Draw a circuit diagram. (c) Find the actual power delivered to the lightbulb in this circuit.

Vishal Gupta
Vishal Gupta
Numerade Educator
05:31

Problem 45

The ammeter shown in Figure $\mathrm{P} 21.45$ reads 2.00 A. Find $I_{1}, I_{2},$ and $\varepsilon$.

Andrija Isakov
Andrija Isakov
Numerade Educator
07:24

Problem 46

The following equations describe an electric circuit: $-I_{1}(220 \Omega)+5.80 \mathrm{V}-I_{2}(370 \Omega)=0$
$+I_{2}(370 \Omega)+I_{3}(150 \Omega)-3.10 \mathrm{V}=0$ $I_{1}+I_{3}-I_{2}=0$
(a) Draw a diagram of the circuit. (b) Calculate the unknowns and identify the physical meaning of each unknown.

Sophie S
Sophie S
Numerade Educator
06:23

Problem 47

The circuit shown in Figure $\mathrm{P} 21.47$ is connected for 2.00 $\min . \quad$ (a) Determine the current in each branch of the circuit. (b) Find the energy delivered by each battery. (c) Find the energy delivered to each resistor. (d) Identify the type of energy storage transformation that occurs in the operation of the circuit. (e) Find the total amount of energy transformed into internal energy in the resistors.

Dominador Tan
Dominador Tan
Numerade Educator
01:48

Problem 48

In Figure $\mathrm{P} 21.47$, show how to add just enough ammeters to measure every different current. Show how to add just enough voltmeters to measure the potential difference across each resistor and across each battery.

Sophie S
Sophie S
Numerade Educator
09:23

Problem 49

Taking $R=1.00 \mathrm{k} \Omega$ and $\varepsilon=250 \mathrm{V}$ in Figure $\mathrm{P} 21.49$, determine the direction and magnitude of the current in the horizontal wire between $a$ and $e$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:15

Problem 50

For the circuit shown in Figure $P 21.50,$ we wish to find the currents $I_{1}, I_{2}$, and $I_{3} .$ Use Kirchhoffs rules to obtain equations for (a) the upper loop, (b) the lower loop, and (c) the junction on the left side. In each case, suppress units for clarity and simplify, combining the terms. (d) Solve the junction equation for $I_{3}$. (e) Using the equation found in part (d), eliminate $I_{3}$ from the equation found in part (b). (f) Solve the equations found in parts (a) and (e) simultaneously for the two unknowns $I_{1}$ and $I_{2}$. (g) Substitute the answers found in part (f) into the junction equation found in part (d), solving for $I_{3}$. (h) What is the significance of the negative answer for $I_{2}$ ?

Dominador Tan
Dominador Tan
Numerade Educator
00:06

Problem 51

In the circuit of Figure $\mathrm{P} 21.51$, determine (a) the current in each resistor and (b) the potential difference across the $200-\Omega$ resistor.

Andrija Isakov
Andrija Isakov
Numerade Educator
05:57

Problem 52

Jumper cables are connected from a fresh battery in one car to charge a dead battery in another car. Figure $\quad$ P21.52 shows the circuit diagram for this situation. While the cables are connected, the ignition switch of the car with the dead battery is closed and the starter is activated to start the engine. Determine the current in
(a) the starter and (b) the dead battery. (c) Is the dead battery being charged while the starter is operating?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:13

Problem 53

Consider a series $R C$ circuit as in Figure $\mathrm{P} 21.53$ for which $R=1.00 \mathrm{M} \Omega$, $C=5.00 \mu \mathrm{F},$ and $\varepsilon=30.0 \mathrm{V}$. Find (a) the time constant of the circuit and (b) the maximum charge on the capacitor after the switch is thrown closed. (c) Find the current in the resistor 10.0 s after the switch is closed.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:39

Problem 54

In places such as hospital operating rooms or factories for electronic circuit boards, electric sparks must be avoided. A person standing on a grounded floor and touching nothing else can typically have a body capacitance of $150 \mathrm{pF}$, in parallel with a foot capacitance of 80.0 pF produced by the dielectric soles of his or her shoes. The person acquires static electric charge from interactions with his or her surroundings. The static charge flows to ground through the equivalent resistance of the two shoe soles in parallel with each other. A pair of rubber-soled street shoes can present an equivalent resistance of $5.00 \times 10^{3}$ M\Omega. A pair of shoes with special static-dissipative soles can have an equivalent resistance of $1.00 \mathrm{M} \Omega$. Consider the person's body and shoes as forming an $R C$ circuit with the ground. (a) How long does it take the rubber-soled shoes to reduce a person's potential from $3.00 \times 10^{3} \mathrm{V}$ to $100 \mathrm{V}$ ? (b) How long does it take the static-dissipative shoes to do the same thing?

Dominador Tan
Dominador Tan
Numerade Educator
03:25

Problem 55

A 2.00 -nF capacitor with an initial charge of $5.10 \mu \mathrm{C}$ is discharged through a 1.30 -k\Omega resistor. (a) Calculate the current in the resistor $9.00 \mu$ s after the resistor is connected across the terminals of the capacitor. (b) What charge remains on the capacitor after $8.00 \mu$ s? (c) What is the maximum current in the resistor?

Sophie S
Sophie S
Numerade Educator
02:31

Problem 56

A $10.0-\mu \mathrm{F}$ capacitor is charged by a 10.0 -V battery through a resistance $R$. The capacitor reaches a potential difference of $4.00 \mathrm{V}$ in a time interval of $3.00 \mathrm{s}$ after charging begins. Find $R$.

Sophie S
Sophie S
Numerade Educator
07:26

Problem 57

In the circuit of Figure $\mathrm{P} 21.57$, the switch $\mathrm{S}$ has been open for a long time. It is then suddenly closed. Take $\varepsilon=10.0 \mathrm{V}$, $R_{1}=50.0 \mathrm{k} \Omega, R_{2}=100 \mathrm{k} \Omega,$ and $C=10.0 \mu \mathrm{F} .$ Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at $t=0 .$ Determine the current in the switch as a function of time.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:33

Problem 58

In the circuit of Figure $\mathrm{P} 21.57$, the switch $\mathrm{S}$ has been open for a long time. It is then suddenly closed. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at $t=0 .$ Determine the current in the switch as a function of time.

Sophie S
Sophie S
Numerade Educator
09:08

Problem 59

The circuit in Figure $\mathrm{P} 21.59$ has been connected for a long time. (a) What is the potential difference across the capacitor? (b) If the battery is disconnected from the circuit, over what time interval does the capacitor discharge to onetenth its initial voltage?

Vishal Gupta
Vishal Gupta
Numerade Educator
01:00

Problem 60

Assume that global lightning on the Earth constitutes a constant current of 1.00 kA between the ground and an atmospheric layer at potential $300 \mathrm{kV}$. (a) Find the power of terrestrial lightning. (b) For comparison, find the power of sunlight falling on the Earth. Sunlight has an intensity of $1370 \mathrm{W} / \mathrm{m}^{2}$ above the atmosphere. Sunlight falls perpendicularly on the circular projected area that the Earth presents to the Sun.

Dominador Tan
Dominador Tan
Numerade Educator
01:07

Problem 61

A current density of $6.00 \times 10^{-13} \mathrm{A} / \mathrm{m}^{2}$ exists in the atmosphere at a location where the electric field is $100 \mathrm{V} / \mathrm{m}$. Calculate the electrical conductivity of the Earth's atmosphere in this region.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
09:44

Problem 62

Lightbulb A is marked "25 W 120 V'," and lightbulb B is marked "100 W 120 V." These labels mean that each lightbulb has its respective power delivered to it when it is connected to a constant 120 -V source. (a) Find the resistance of each lightbulb. (b) During what time interval does $1.00 \mathrm{C}$ pass into lightbulb A? (c) Is this charge different upon its exit versus its entry into the lightbulb? Explain. (d) In what time interval does $1.00 \mathrm{J}$ pass into lightbulb $\mathrm{A}$ ? (e) By what mechanisms does this energy enter and exit the lightbulb? Explain. (f) Find the cost of running lightbulb A continuously for 30.0 days, assuming the electric company sells its product at $\$ 0.110$ per kWh.

Jonathan Ibarra
Jonathan Ibarra
Numerade Educator
02:21

Problem 63

A straight, cylindrical wire lying along the $x$ axis has a length of $0.500 \mathrm{m}$ and a diameter of $0.200 \mathrm{mm} .$ It is made of a material described by Ohm's law with a resistivity of $\rho=$ $4.00 \times 10^{-8} \Omega \cdot \mathrm{m} .$ Assume a potential of $4.00 \mathrm{V}$ is maintained at the left end of the wire at $x=0 .$ Also assume $V=0$at $x=0.500 \mathrm{m} .$ Find (a) the magnitude and direction of the electric field in the wire, (b) the resistance of the wire, (c) the magnitude and direction of the electric current in the wire, and (d) the current density in the wire. (e) Show that $E=\rho J$.

Kayla Gephart
Kayla Gephart
Numerade Educator
02:28

Problem 64

A straight, cylindrical wire lying along the $x$ axis has a length $L$ and a diameter $d$. It is made of a material described by Ohm's law with a resistivity $\rho$. Assume potential $V$ is maintained at the left end of the wire at $x=0 .$ Also assume the potential is zero at $x=L .$ In terms of $L, d, V, \rho,$ and physical constants, derive expressions for (a) the magnitude and direction of the electric field in the wire, (b) the resistance of the wire, (c) the magnitude and direction of the electric current in the wire, and (d) the current density in the wire.
(e) Show that $E=\rho J$.

Kayla Gephart
Kayla Gephart
Numerade Educator
01:16

Problem 65

Four 1.50 -V AA batteries in series are used to power a small radio. If the batteries can move a charge of $240 \mathrm{C}$, how long will they last if the radio has a resistance of $200 \Omega$ ?

Narayan Hari
Narayan Hari
Numerade Educator
03:21

Problem 66

An oceanographer is studying how the ion concentration in seawater depends on depth. She makes a measurement by lowering into the water a pair of concentric metallic cylinders (Fig. $\mathrm{P} 21.66$ ) at the end of a cable and taking data to determine the resistance between these electrodes as a function of depth. The water between the two cylinders forms a cylindrical shell of inner radius $r_{a},$ outer radius $r_{b},$ and length $L$ much larger than $r_{b} .$ The scientist applies a potential difference $\Delta V$ between the inner and outer surfaces, producing an outward radial current I. Let $\rho$ represent the resistivity of the water. (a) Find the resistance of the water between the cylinders in terms of $L, \rho, r_{a},$ an $r_{b} .$ (b) Express the resistivity of the water in terms of the measured quantities $L, r_{a}, r_{b}, \Delta V,$ and $I$.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
06:24

Problem 67

The values of the components in a simple series $R C$ circuit containing a switch (Fig. P21.53) are $C=1.00 \mu \mathrm{F}$, $R=2.00 \times 10^{6} \Omega,$ and $\varepsilon=10.0 \mathrm{V} .$ At the instant $10.0 \mathrm{s}$
after the switch is closed, calculate (a) the charge on the capacitor, (b) the current in the resistor, (c) the rate at which energy is being stored in the capacitor, and (d) the rate at which energy is being delivered by the battery.

Keshav Singh
Keshav Singh
Numerade Educator
07:51

Problem 68

A battery is used to charge a capacitor through a resistor as shown in Figure $\mathrm{P} 21.53 .$ Show that half the energy supplied by the battery appears as internal energy in the resistor and half is stored in the capacitor.

Artemisa Mazón
Artemisa Mazón
Numerade Educator
02:51

Problem 69

Switch S shown in Figure P21.69 has been closed for a long time, and the electric circuit carries a constant current. Take $C_{1}=3.00 \mu \mathrm{F}, C_{2}=6.00 \mu \mathrm{F}$, $R_{1}=4.00 \mathrm{k} \Omega,$ and $R_{2}=$
$7.00 \mathrm{k} \Omega .$ The power delivered to $R$, is $2.40 \mathrm{W}$. (a) Find the charge on $C_{1}$. (b) Now the switch is opened. After many milliseconds, by how much has the charge on $C_{2}$ changed?

Dominador Tan
Dominador Tan
Numerade Educator
04:08

Problem 70

Why is the following situation impossible? A battery has an emf of $\varepsilon=9.20 \mathrm{V}$ and an internal resistance of $r=1.20 \Omega$. A resistance $R$ is connected across the battery and extracts from it a power of $P=21.2 \mathrm{W}$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:52

Problem 71

The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P21.71). The unknown resistance $R_{x}$ is between points $C$ and $E .$ Point $E$ is a true ground, but it is inaccessible for direct measurement because this stratum is several meters below the Earth's surface. Two identical rods are driven into the ground at $A$ and $B$, introducing an unknown resistance $R_{y}$. The procedure is as follows. Measure resistance $R_{1}$ between points $A$ and $B,$ then connect $A$ and $B$ with a heavy conducting wire and measure resistance $R_{2}$ between points $A$ and $C .$ (a) Derive an equation for $R_{x}$ in terms of the observable resistances, $R_{1}$ and $R_{2} .$ (b) A satisfactory ground resistance would be $R_{x}<2.00 \Omega$. Is the grounding of the station adequate if measurements give $R_{1}=13.0 \Omega$ and $R_{2}=6.00 \Omega ?$ Explain.

Keshav Singh
Keshav Singh
Numerade Educator
02:45

Problem 72

The circuit shown in Figure $\mathrm{P} 21.72$ is set up in the laboratory to measure an unknown capacitance $C$ in series with a resistance $R=$ $10.0 \mathrm{M} \Omega$ powered by a battery whose emf is $6.19 \mathrm{V}$. The data given in the table are the measured voltages across the capacitor as a function of time, where $t=0$ represents the instant at which the switch is thrown to position $b .$ (a) Construct a graph of $\ln (\mathcal{E} / \Delta V)$ versus $t$ and perform a linear least-squares fit to the data. (b) From the slope of your graph, obtain a value for the time constant of the circuit and a value for the capacitance.

Dominador Tan
Dominador Tan
Numerade Educator
01:49

Problem 73

A battery has an emf $\varepsilon$ and internal resistance $r$. A variable load resistor $R$ is connected across the terminals of the battery. (a) Determine the value of $R$ such that the potential difference across the terminals is a maximum. (b) Determine the value of $R$ so that the current in the circuit is a maximum. (c) Determine the value of $R$ so that the power delivered to the load resistor is a maximum. Choosing the load resistance for maximum power transfer is a case of what is called impedance matching in general. Impedance matching is important in shifting gears on a bicycle, in connecting a loudspeaker to an audio amplifier, in connecting a battery charger to a bank of solar photoelectric cells, and in many other applications.

Dominador Tan
Dominador Tan
Numerade Educator
06:33

Problem 74

The switch in Figure $\mathrm{P} 21.74$ a closes when $\Delta V_{c}>\frac{2}{3} \Delta V$ and opens when $\Delta V_{c}<\frac{1}{3} \Delta V .$ The ideal voltmeter reads a potential difference as plotted in Figure $\mathrm{P} 21.74 \mathrm{b}$. What is the period $T$ of the waveform in terms of $R_{1}, R_{2},$ and $C ?$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:21

Problem 75

An electric heater is rated at $1.50 \times 10^{3} \mathrm{W}$, a toaster at $750 \mathrm{W}$, and an electric grill at $1.00 \times 10^{3} \mathrm{W}$. The three appliances are connected to a common 120-V household circuit.
(a) How much current does each draw? (b) If the circuit is protected with a 25.0 -A circuit breaker, will the circuit breaker be tripped in this situation? Explain your answer.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:45

Problem 76

An experiment is conducted to measure the electrical resistivity of Nichrome in the form of wires with different lengths and cross-sectional areas. For one set of measurements, a student uses 30 -gauge wire, which has a cross-sectional area of $7.30 \times 10^{-8} \mathrm{m}^{2} .$ The student measures the potential difference across the wire and the current in the wire with a voltmeter and an ammeter, respectively. For each set of measurements given in the table taken on wires of three different lengths, calculate the resistance of the wires and the corresponding values of the resistivity. (b) What is the average value of the resistivity? (c) Explain how this value compares with the value given in Table 21.1.

Dominador Tan
Dominador Tan
Numerade Educator
07:43

Problem 77

Four resistors are connected in parallel across a $9.20-\mathrm{V}$ battery. They carry currents of $150 \mathrm{mA}, 45.0 \mathrm{mA}, 14.0 \mathrm{mA}$ and $4.00 \mathrm{mA}$. If the resistor with the largest resistance is replaced with one having twice the resistance, (a) what is the ratio of the new current in the battery to the original current? (b) What If? If instead the resistor with the smallest resistance is replaced with one having twice the resistance, what is the ratio of the new total current to the original current? $(\mathrm{c})$ On a February night, energy leaves a house by several energy leaks, including $1.50 \times 10^{3} \mathrm{W}$ by conduction through the ceiling, $450 \mathrm{W}$ by infiltration (air-flow) around the windows, $140 \mathrm{W}$ by conduction through the basement wall above the foundation sill, and $40.0 \mathrm{W}$ by conduction through the plywood door to the attic. To produce the biggest saving in heating bills, which one of these energy transfers should be reduced first? Explain how you decide. Clifford Swartz suggested the idea for this problem.

Sheh Lit Chang
Sheh Lit Chang
University of Washington